Stevens, Jan
Non-embeddable 1-convex manifolds  [ Variétés 1-convexes non plongeables ]
Annales de l'institut Fourier, Tome 64 (2014) no. 5 , p. 2205-2222
MR 3330936 | Zbl 06387336
doi : 10.5802/aif.2909
URL stable : http://www.numdam.org/item?id=AIF_2014__64_5_2205_0

Classification:  32S45,  32F10,  32Q15,  32T15,  13C20,  14E30
Mots clés: variétés 1-convexes, petites résolutions
Nous montrons que chaque petite résolution d’une singularité de hypersurface 3-dimensionnelle terminale peut se produire sur une variété 1-convexe non plongeable. Nous donnons un exemple explicite d’une variété non plongeable contenant une courbe exceptionnelle rationnelle irréductible avec fibré normal du type (1,-3). À cette fin, nous étudions de petites résolutions des singularités cD 4 .
We show that every small resolution of a 3-dimensional terminal hypersurface singularity can occur on a non-embeddable 1-convex manifold. We give an explicit example of a non-embeddable manifold containing an irreducible exceptional rational curve with normal bundle of type (1,-3). To this end we study small resolutions of cD 4 -singularities.

Bibliographie

[1] Alessandrini, L.; Bassanelli, G. On the embedding of 1-convex manifolds with 1-dimensional exceptional set, Ann. Inst. Fourier (Grenoble), 51 (2001) no. 1, p. 99–108 Article  Numdam | MR 1821070 | Zbl 0966.32008

[2] Alessandrini, Lucia; Bassanelli, Giovanni Transforms of currents by modifications and 1-convex manifolds, Osaka J. Math., 40 (2003) no. 3, p. 717–740 MR 2003745 | Zbl 1034.32009

[3] Arnol’D, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. I, Birkhäuser Boston, Inc., Boston, MA, Monographs in Mathematics, 82 (1985), p. xi+382 (The classification of critical points, caustics and wave fronts, Translated from the Russian by Ian Porteous and Mark Reynolds) MR 777682 | Zbl 0554.58001

[4] Bassanelli, Giovanni; Leoni, Marco Some examples of 1-convex non-embeddable threefolds, Rev. Roumaine Math. Pures Appl., 52 (2007) no. 6, p. 611–617 MR 2387599 | Zbl 1174.32014

[5] Clemens, Herbert; Kollár, János; Mori, Shigefumi Higher-dimensional complex geometry (Astérisque 166, (1988), 144 pp.) MR 1004926 | Zbl 0689.14016

[6] Colţoiu, Mihnea On 1-convex manifolds with 1-dimensional exceptional set, Rev. Roumaine Math. Pures Appl., 43 (1998) no. 1-2, p. 97–104 (Collection of papers in memory of Martin Jurchescu) MR 1655264 | Zbl 0932.32018

[7] Colţoiu, Mihnea Some remarks about 1-convex manifolds on which all holomorphic line bundles are trivial, Bull. Sci. Math., 130 (2006) no. 4, p. 337–340 Article  MR 2237448 | Zbl 1111.32007

[8] Katz, Sheldon; Morrison, David R. Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Algebraic Geom., 1 (1992) no. 3, p. 449–530 MR 1158626 | Zbl 0788.14036

[9] Kawamata, Yujiro General hyperplane sections of nonsingular flops in dimension 3, Math. Res. Lett., 1 (1994) no. 1, p. 49–52 Article  MR 1258489 | Zbl 0834.32007

[10] Kollár, János Flips, flops, minimal models, etc, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA (1991), p. 113–199 MR 1144527 | Zbl 0755.14003

[11] Laufer, Henry B. On CP 1 as an exceptional set, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979), Princeton Univ. Press, Princeton, N.J. (Ann. of Math. Stud.) 100 (1981), p. 261–275 MR 627762 | Zbl 0523.32007

[12] Moĭšezon, B. G. Irreducible exceptional submanifolds, of the first kind, of three-dimensional complex-analytic manifolds, Soviet Math. Dokl., 6 (1965), p. 402–403 MR 222916 | Zbl 0158.33103

[13] Pinkham, Henry C. Factorization of birational maps in dimension 3, Singularities, Part 2 (Arcata, Calif., 1981), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) 40 (1983), p. 343–371 MR 713260 | Zbl 0544.14005

[14] Ravindra, G. V.; Srinivas, V. The Grothendieck-Lefschetz theorem for normal projective varieties, J. Algebraic Geom., 15 (2006) no. 3, p. 563–590 Article  MR 2219849 | Zbl 1123.14004

[15] Ravindra, G. V.; Srinivas, V. The Noether-Lefschetz theorem for the divisor class group, J. Algebra, 322 (2009) no. 9, p. 3373–3391 Article  MR 2567426 | Zbl 1189.14010

[16] Reid, Miles Minimal models of canonical 3-folds, Algebraic varieties and analytic varieties (Tokyo, 1981), North-Holland, Amsterdam (Adv. Stud. Pure Math.) 1 (1983), p. 131–180 MR 715649 | Zbl 0558.14028

[17] Schneider, Michael Familien negativer Vektorraumbündel und 1-konvexe Abbildungen, Abh. Math. Sem. Univ. Hamburg, 47 (1978), p. 150–170 (Special issue dedicated to the seventieth birthday of Erich Kähler) Article  MR 492393 | Zbl 0391.32011

[18] Tan, Vo Van On certain non-Kählerian strongly pseudoconvex manifolds, J. Geom. Anal., 4 (1994) no. 2, p. 233–245 Article  MR 1277508 | Zbl 0807.32018

[19] Tan, Vo Van On the Kählerian geometry of 1-convex threefolds, Forum Math., 7 (1995) no. 2, p. 131–146 MR 1316945 | Zbl 0839.32003

[20] Tjurina, G. N. Resolution of singularities of flat deformations of double rational points, Funkcional. Anal. i Priložen., 4 (1970) no. 1, p. 77–83 MR 267129 | Zbl 0221.32008

[21] Vâjâitu, Viorel On embeddable 1-convex spaces, Osaka J. Math., 38 (2001) no. 2, p. 287–294 MR 1833621 | Zbl 0982.32010

[22] Vo Van, Tan On the quasi-projectivity of compactifiable strongly pseudoconvex manifolds, Bull. Sci. Math., 129 (2005) no. 6, p. 501–522 Article  MR 2142895 | Zbl 1083.32010